We address the practical problems of estimating the information relations that characterize large networks. Building on methods developed for analysis of the neural code, we show that reliable estimates of mutual information can be obtained with manageable computational effort. The same methods allow estimation of higher order, multi--information terms. These ideas are illustrated by analyses of gene expression, financial markets, and consumer preferences. In each case, information theoretic measures correlate with independent, intuitive measures of the underlying structures in the system.
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Washington（华盛顿大学）
Estimating mutual information from observed samples is a basic primitive, useful in several machine learning tasks including correlation mining, information bottleneck clustering, learning a Chow-Liu tree, and conditional independence testing in (causal) graphical models. While mutual information is a well-defined quantity in general probability spaces, existing estimators can only handle two special cases of purely discrete or purely continuous pairs of random variables. The main challenge is that these methods first estimate the (differential) entropies of X, Y and the pair (X;Y) and add the…
Estimating mutual information (MI) from samples is a fundamental problem in statistics, machine learning, and data analysis. Recently it was shown that a popular class of non-parametric MI estimators perform very poorly for strongly dependent variables and have sample complexity that scales exponentially with the true MI. This undesired behavior was attributed to the reliance of those estimators on local uniformity of the underlying (and unknown) probability density function. Here we present a novel semi-parametric estimator of mutual information, where at each sample point, densities are {\em…
Estimating mutual information (MI) between two continuous random variables $X$ and $Y$ allows to capture non-linear dependencies between them, non-parametrically. As such, MI estimation lies at the core of many data science applications. Yet, robustly estimating MI for high-dimensional $X$ and $Y$ is still an open research question. In this paper, we formulate this problem through the lens of manifold learning. That is, we leverage the common assumption that the information of $X$ and $Y$ is captured by a low-dimensional manifold embedded in the observed high-dimensional space and transfer it …
Estimating conditional mutual information (CMI) is an essential yet challenging step in many machine learning and data mining tasks. Estimating CMI from data that contains both discrete and continuous variables, or even discrete-continuous mixture variables, is a particularly hard problem. In this paper, we show that CMI for such mixture variables, defined based on the Radon-Nikodym derivate, can be written as a sum of entropies, just like CMI for purely discrete or continuous data. Further, we show that CMI can be consistently estimated for discrete-continuous mixture variables by learning an…
Multivariate pattern analyses approaches in neuroimaging are fundamentally concerned with investigating the quantity and type of information processed by various regions of the human brain; typically, estimates of classification accuracy are used to quantify information. While a extensive and powerful library of methods can be applied to train and assess classifiers, it is not always clear how to use the resulting measures of classification performance to draw scientific conclusions: e.g. for the purpose of evaluating redundancy between brain regions. An additional confound for interpreting cl…
Total correlation (TC) is a fundamental concept in information theory that measures statistical dependency among multiple random variables. Recently, TC has shown noticeable effectiveness as a regularizer in many learning tasks, where the correlation among multiple latent embeddings requires to be jointly minimized or maximized. However, calculating precise TC values is challenging, especially when the closed-form distributions of embedding variables are unknown. In this paper, we introduce a unified framework to estimate total correlation values with sample-based mutual information (MI) estim…
In many real-world problems, we want to infer some property of an expensive black-box function $f$, given a budget of $T$ function evaluations. One example is budget constrained global optimization of $f$, for which Bayesian optimization is a popular method. Other properties of interest include local optima, level sets, integrals, or graph-structured information induced by $f$. Often, we can find an algorithm $\mathcal{A}$ to compute the desired property, but it may require far more than $T$ queries to execute. Given such an $\mathcal{A}$, and a prior distribution over $f$, we refer to the pro…
We consider the processing of statistical samples $X\sim P_θ$ by a channel $p(y|x)$, and characterize how the statistical information from the samples for estimating the parameter $θ\in\mathbb{R}^d$ can scale with the mutual information or capacity of the channel. We show that if the statistical model has a sub-Gaussian score function, then the trace of the Fisher information matrix for estimating $θ$ from $Y$ can scale at most linearly with the mutual information between $X$ and $Y$. We apply this result to obtain minimax lower bounds in distributed statistical estimation problems, and obtain…
The estimation of mutual information (MI) or conditional mutual information (CMI) from a set of samples is a long-standing problem. A recent line of work in this area has leveraged the approximation power of artificial neural networks and has shown improvements over conventional methods. One important challenge in this new approach is the need to obtain, given the original dataset, a different set where the samples are distributed according to a specific product density function. This is particularly challenging when estimating CMI. In this paper, we introduce a new technique, based on k neare…
We propose a new estimator to measure directed dependencies in time series. The dimensionality of data is first reduced using a new non-uniform embedding technique, where the variables are ranked according to a weighted sum of the amount of new information and improvement of the prediction accuracy provided by the variables. Then, using a greedy approach, the most informative subsets are selected in an iterative way. The algorithm terminates, when the highest ranked variable is not able to significantly improve the accuracy of the prediction as compared to that obtained using the existing sele…
CRYPTO-MINE: Cryptanalysis via Mutual Information Neural Estimation
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Massachusetts Institute of Technology（麻省理工学院）；Technion（以色列理工学院）；Clemson University（克莱姆森大学）；MIT Lincoln Laboratory（麻省理工学院林肯实验室）
Let $X$, $Y$ be two correlated discrete random variables. We consider an estimation of $X$ from encoded data $φ(Y)$ of $Y$ by some encoder function $φ(Y)$. We derive an inequality describing a relation of the correct probability of estimation and the mutual information between $X$ and $φ(Y)$. This inequality may be useful for the secure analysis of crypto system when we use the success probability of estimating secret data as a security criterion. It also provides an intuitive meaning of the secrecy exponent in the strong secrecy criterion.
Recently, several methods have been proposed for estimating the mutual information from sample data using deep neural networks and without the knowing closed form distribution of the data. This class of estimators is referred to as neural mutual information estimators. Although very promising, such techniques have yet to be rigorously bench-marked so as to establish their efficacy, ease of implementation, and stability for capacity estimation which is joint maximization frame-work. In this paper, we compare the different techniques proposed in the literature for estimating capacity and provide…
We consider the problem of estimating mutual information between dependent data, an important problem in many science and engineering applications. We propose a data-driven, non-parametric estimator of mutual information in this paper. The main novelty of our solution lies in transforming the data to frequency domain to make the problem tractable. We define a novel metric--mutual information in frequency--to detect and quantify the dependence between two random processes across frequency using Cramér's spectral representation. Our solution calculates mutual information as a function of frequen…
Variational approaches based on neural networks are showing promise for estimating mutual information (MI) between high dimensional variables. However, they can be difficult to use in practice due to poorly understood bias/variance tradeoffs. We theoretically show that, under some conditions, estimators such as MINE exhibit variance that could grow exponentially with the true amount of underlying MI. We also empirically demonstrate that existing estimators fail to satisfy basic self-consistency properties of MI, such as data processing and additivity under independence. Based on a unified pers…
Applied Artificial Intelligence Institute, Deakin University（迪肯大学应用人工智能研究所）
Mutual Information (MI) and Conditional Mutual Information (CMI) are multi-purpose tools from information theory that are able to naturally measure the statistical dependencies between random variables, thus they are usually of central interest in several statistical and machine learning tasks, such as conditional independence testing and representation learning. However, estimating CMI, or even MI, is infamously challenging due the intractable formulation. In this study, we introduce DINE (Diffeomorphic Information Neural Estimator)-a novel approach for estimating CMI of continuous random var…
Hitachi (China) Research & Development Corporation（日立（中国）研究开发有限公司）
Causal discovery is a fundamental problem in statistics and has wide applications in different fields. Transfer Entropy (TE) is a important notion defined for measuring causality, which is essentially conditional Mutual Information (MI). Copula Entropy (CE) is a theory on measurement of statistical independence and is equivalent to MI. In this paper, we prove that TE can be represented with only CE and then propose a non-parametric method for estimating TE via CE. The proposed method was applied to analyze the Beijing PM2.5 data in the experiments. Experimental results show that the proposed m…
In the era of big data and the Internet of Things (IoT), data owners need to share a large amount of data with the intended receivers in an insecure environment, posing a trade-off issue between user privacy and data utility. The privacy utility trade-off was facilitated through a privacy funnel based on mutual information. Nevertheless, it is challenging to characterize the mutual information accurately with small sample size or unknown distribution functions. In this article, we propose a privacy funnel based on mutual information neural estimator (MINE) to optimize the privacy utility trade…
We present a method to generate speech from input text and a style vector that is extracted from a reference speech signal in an unsupervised manner, i.e., no style annotation, such as speaker information, is required. Existing unsupervised methods, during training, generate speech by computing style from the corresponding ground truth sample and use a decoder to combine the style vector with the input text. Training the model in such a way leaks content information into the style vector. The decoder can use the leaked content and ignore some of the input text to minimize the reconstruction lo…
In recent years, there has been an upswing of interest in estimating information from data emerging in a lot of areas beyond communications. This paper aims at estimating the information between two random phenomena by using consolidated second-order statistics tools. The squared-loss mutual information is chosen for that purpose as a natural surrogate of Shannon mutual information. The rationale for doing so is developed for i.i.d. discrete sources -mapping data onto the simplex space-, and for analog sources -mapping data onto the characteristic space-, highlighting the links with other well…
We review the current literature concerned with information plane analyses of neural network classifiers. While the underlying information bottleneck theory and the claim that information-theoretic compression is causally linked to generalization are plausible, empirical evidence was found to be both supporting and conflicting. We review this evidence together with a detailed analysis of how the respective information quantities were estimated. Our survey suggests that compression visualized in information planes is not necessarily information-theoretic, but is rather often compatible with geo…
The integration and transfer of information from multiple sources to multiple targets is a core motive of neural systems. The emerging field of partial information decomposition (PID) provides a novel information-theoretic lens into these mechanisms by identifying synergistic, redundant, and unique contributions to the mutual information between one and several variables. While many works have studied aspects of PID for Gaussian and discrete distributions, the case of general continuous distributions is still uncharted territory. In this work we present a method for estimating the unique infor…
Recent works investigated the generalization properties in deep neural networks (DNNs) by studying the Information Bottleneck in DNNs. However, the mea- surement of the mutual information (MI) is often inaccurate due to the density estimation. To address this issue, we propose to measure the dependency instead of MI between layers in DNNs. Specifically, we propose to use Hilbert-Schmidt Independence Criterion (HSIC) as the dependency measure, which can measure the dependence of two random variables without estimating probability densities. Moreover, HSIC is a special case of the Squared-loss M…
Recently, several methods have been proposed for estimating the mutual information from sample data using deep neural networks. These estimators ar referred to as neural mutual information estimation (NMIE)s. NMIEs differ from other approaches as they are data-driven estimators. As such, they have the potential to perform well on a large class of capacity problems. In order to test the performance across various NMIEs, it is desirable to establish a benchmark encompassing the different challenges of capacity estimation. This is the objective of this paper. In particular, we consider three scen…
We investigate a multi-user multiple-input multiple-output interference network in the presence of multiple reconfigurable intelligent surfaces (RISs). The entire system is described by using a circuit-based model for the transmitters, receivers, and RISs. This is obtained by leveraging the electromagnetic tool of mutual impedances, which accounts for the signal propagation and the mutual coupling among closely-spaced scattering elements. An iterative and provably convergent optimization algorithm that maximizes the sum-rate of RIS-assisted multi-user interference channels is introduced. Numer…
Estimating the entropy of a discrete random variable is a fundamental problem in information theory and related fields. This problem has many applications in various domains, including machine learning, statistics and data compression. Over the years, a variety of estimation schemes have been suggested. However, despite significant progress, most methods still struggle when the sample is small, compared to the variable's alphabet size. In this work, we introduce a practical solution to this problem, which extends the work of McAllester and Statos (2020). The proposed scheme uses the generaliza…
We investigate the sets of joint probability distributions that maximize the average multi-information over a collection of margins. These functionals serve as proxies for maximizing the multi-information of a set of variables or the mutual information of two subsets of variables, at a lower computation and estimation complexity. We describe the maximizers and their relations to the maximizers of the multi-information and the mutual information.
Estimating entropy and mutual information consistently is important for many machine learning applications. The Kozachenko-Leonenko (KL) estimator (Kozachenko & Leonenko, 1987) is a widely used nonparametric estimator for the entropy of multivariate continuous random variables, as well as the basis of the mutual information estimator of Kraskov et al. (2004), perhaps the most widely used estimator of mutual information in this setting. Despite the practical importance of these estimators, major theoretical questions regarding their finite-sample behavior remain open. This paper proves finite-s…
We propose a novel estimator of the mutual information between two ordinal vectors $x$ and $y$. Our approach is inductive (as opposed to deductive) in that it depends on the data generating distribution solely through some nonparametric properties revealing associations in the data, and does not require having enough data to fully characterize the true joint distributions $P_{x, y}$. Specifically, our approach consists of (i) noting that $I\left(y; x\right) = I\left(u_y; u_x\right)$ where $u_y$ and $u_x$ are the copula-uniform dual representations of $y$ and $x$ (i.e. their images under the pr…
